Question: Simplify the following expression and state the condition under which the simplification is valid: $t = \dfrac{y^2 - y - 2}{y^2 + 2y - 8}$
First factor the expressions in the numerator and denominator. $ \dfrac{y^2 - y - 2}{y^2 + 2y - 8} = \dfrac{(y + 1)(y - 2)}{(y + 4)(y - 2)} $ Notice that the term $(y - 2)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(y - 2)$ gives: $t = \dfrac{y + 1}{y + 4}$ Since we divided by $(y - 2)$, $y \neq 2$. $t = \dfrac{y + 1}{y + 4}; \space y \neq 2$